On Designs of Maximal (+1, -1)-Matrices of Order n ≡2 (mod 4). II

0(A) = AAT = (an), G(B) = BB* = (6,-,-), 1 Û i, j Û n/2; where an = í>,-¿ = n/2, for 1 ^ i ^ n/2, and a<y + fc.y = 2 for i í¿ j. Since A, 5 are circulant (+1, — l)-matrices, it can be shown easily that G(A), G(B) are not only circulant but also symmetric, namely, o,-,= 0|<_,-| and bu = b\i-j\. It follows that construction of Mn is reduced to finding two finite sequences [ak] and {bk\, 1 Sí k ¿ (n — 2)/4, such that ak + bk = 2. Let C = (cy) be an mth order circulant (+1, — l)-matrix, then G(C) = G(CT) = G(Cpq), where Cpq = (cki), k = p + i, I = q + j (mod m) for fixed integers p and q. Consequently, the finite sequences of C, CT and Cpq are identical; therefore, matrices C, CT and Cpq are regarded as of the same type. In the following table, all M„, constructible by all distinct types of A and B with the restriction that N(A) ^ N(B) < n/A, where N(C) means the number of — l's in each row of C, are listed for n ^ 38. The following methods and theorems are helpful for constructions of M„. Let S = (s,y) be the mth order circulant matrix such that